Wrapping the Cube and other Polyhedra

T. Tarnai, F. KovacsDepartment of Structural Mechanics,

Budapest University of Technology and Economics,Budapest, Muegyetem rkp. 3., H-1521 Hungary

P.W. FowlerDepartment of Chemistry, University of Sheffield,

Sheffield S3 7HF, UKS.D. Guest

Department of Engineering, University of Cambridge,Trumpington Street, Cambridge CB2 1PZ, UK

April 19, 2012

Abstract

An infinite series of 2-fold, 2-way weavings of the cube, correspond-ing to ‘wrappings’, or double covers of the cube, is described with theaid of the 2-parameter Goldberg-Coxeter construction. The strands ofall such wrappings correspond to the central circuits of octahedrites (4-regular polyhedral graphs with square and triangular faces), which forthe cube necessarily have octahedral symmetry. Removing the sym-metry constraint leads to wrappings of other eight-vertex convex poly-hedra. Moreover, wrappings of convex polyhedra with fewer verticescan be generated by generalising from octahedrites to i-hedrites, whichadditionally include digonal faces. When the strands of a wrapping cor-respond to the central circuits of a 4-regular graph that includes facesof size greater than 4, non-convex ‘crinkled’ wrappings are generated.The various generalisations have implications for activities as diverseas the construction of woven closed baskets and the manufacture ofadvanced composite components of complex geometry.

1 Introduction

Carbon-fibre composites are used throughout advanced manufacturing, andfigure in, for instance, the latest aerospace components (Toensmeier, 2005).In many applications, tows (bundles) of fibres are used in the form of aweave (Onal and Adanur, 2007); in other applications, tows of fibres arewrapped on a former, using tow placement machines (Rudd et al., 1999).Directly related to these modern technologies is the long-established weav-ing of baskets, open and closed, a technology common to many cultures and

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Figure 1: Skew weaving of the cube with b = 5 and c = 2 (see text for defi-nitions). The dark ribbon is a single closed strand. The whole weaving has3 symmetrically related strands of length 116 unit squares each. This closedbasket was constructed by Felicity Wood of the Oxfordshire BasketmakersAssociation, who provided the photograph.

periods (Tarnai, 2006; Pitt Rivers Museum, 2009), which continues to gener-ate applications in art and craft (Kavicky, 2004; Pulleyn, 1991) and modernarchitecture (Ministry of Land, Infrastructure and Transport and NihonSekkei Inc., 2005). Closed baskets are often considered as woven spheresor polyhedra, and are treated in many mathematical reviews and books,e.g., Pedersen (1981); Gerdes (1999); von Randow (2004). The present pa-per examines mathematical aspects of weaving on polyhedral surfaces, withpractical applications in mind, concentrating initially on weavings on thecube (Figure 1), before extending the treatment to a class of weavings thatturn out to be described by the ‘octahedrites’ of Deza and Shtogrin (2003).

For the plane, the simplest weaving is the 2-fold, 2-way weave (Grunbaumand Shephard, 1988) in which a typical point is covered by two strands(hence ‘2-fold’), with the strands crossing at right angles (hence ‘2-way’), inan overall check pattern where an individual strand passes alternately overand under at crossings. The same basic definition can be applied to a weav-ing of a closed basket on the surface of the cube (Tarnai, 2006), and, as weshall see, to other polyhedra; as on the plane, strands cross at right angles,and each strand passes alternately over and under at crossings. However, thestrands are now necessarily of finite length, and may have self-intersections.

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For construction and classification, it is convenient to simplify the physicalweave to a double cover, where the up-down relationship of the strands hasbeen ‘flattened out’, so that, apart from points on strand boundaries, ev-ery point belongs to two orthogonal portions of strands, with no conceptof one strand being above another. In what follows, we will be concernedwith the properties of this double-cover version of the weaving, which wecan informally call a ‘wrapping’.

For the particular case of the cube, the strict alternation of the checkpattern is necessarily disrupted at vertices, and the symmetries of the overallpattern are restricted to a subset of those of the underlying cube. Weav-ings of the cube fall into three types, depending on the pairs of angles ofintersection between the strand and the cube edges: Class I (0, π/2), ClassII (π/4, π/4), Class III (θ, π/2 − θ), with 0 < θ < π/4. Examples of allthree are illustrated in Figure 2. This classification echoes the schemes forgeodesic domes (Coxeter, 1971; Wenninger, 1979), and for classifying carbonnanotubes into armchair, zig-zag and chiral types (Hamada et al., 1992). Aswrappings, double covers in Classes I and II have the full set of symmetriesof the cube, whereas those in Class III have only the rotations.

Consideration of the ways in which wrappings of the cube can be rep-resented and classified leads naturally to polyhedral graphs, from which itbecomes apparent that all cube wrappings can be represented as members ofthe family of octahedrite graphs (Deza and Shtogrin, 2003). Generalisations,by removal of the restriction to octahedral symmetry, by addition of digo-nal faces to the octahedrite recipe, or by introduction of general face sizes,will be shown to generate further infinite families of convex and non-convexwrapped polyhedra, and hence of closed baskets.

2 Geometrical approach to cube wrappings

A natural representation of a double covering of the plane by strands showsthe strand boundaries as orthogonal lines. This leads to a tessellation of theplane by square tiles, meeting four at a vertex, i.e., {4, 4} in the Schlafli no-tation (Coxeter, 1969). Each square tile corresponds to two overlaid portionsof strands of the original weaving. Many different weavings may correspondto a given double cover (Grunbaum and Shephard, 1988).

Wrappings of the cube can be represented in a similar way by drawinga net of the cube onto the {4, 4} tessellation, with the restriction that thevertices of the net are lattice points, i.e., points at which strand boundariescross. This restriction follows from the observation that a cube vertex cannotlie within a strand, as the sum of angles at a cube vertex is only 3π/2. Eachsuch net can be described by a symbol {4, 3+}b,c, where the 4 implies atiling by squares, and the 3+ indicates that three or more square tiles meetat each vertex of the net. The integers b and c (b ≥ 0, c ≥ 0, b+c > 0) define

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(a.i) (b.i) (c.i)

(a.ii) (b.ii) (c.ii)

(a.iii) (b.iii) (c.iii)

(a.iv) (b.iv) (c.iv)

topfront

bottom

Figure 2: The three classes of cube wrappings. Columns (a), (b) and (c)illustrate wrappings of class I, II and III respectively. The illustrated wrap-pings are {4, 3+}3,0, {4, 3+}2,2 and {4, 3+}3,1. Row (i) shows a single strandwrapped onto the cube, and row (ii) shows the same strand on the unfoldednet of the cube. Row (iii) shows the complete weaving, with the single strandhighlighted. Row (iv) shows the dual maps of the wrappings as graphs em-bedded on the cube.

4

00

1

1

2 …

…

b

c

Figure 3: Definition of parameters b and c for tiling {4, 3+}b,c of the cube.

how all the (congruent) faces of the net lie on the underlying tessellationof the plane (Figure 3): from any starting vertex, an adjacent vertex isreached by making b steps along edges of the tessellation in one direction,followed by c steps after a change in direction by an angle of π/2. This2-parameter construction is ultimately derived from the work of Goldberg(1937) and Coxeter (1971), and has been applied in the present context byseveral authors (Tarnai, 2006; Dutour and Deza, 2004; Deza and DutourSikiric, 2007)

If a square tile has unit area (if each strand has unit width), the areaof each face of the net is S = b2 + c2, and the total length (area) of allstrands is therefore 12S, and the angle at which a strand meets a cube edgeis tan−1(b/c), or its complement tan−1(c/b). Class I corresponds to the caseb = 0 or c = 0; {4, 3+}b,0 is identical with {4, 3+}0,b. Class II correspondsto the case b = c. Class III corresponds to b 6= 0, c 6= 0, b 6= c, and thepair of Class III wrappings with symbols {4, 3+}b,c and {4, 3+}c,b are enan-tiomeric as a consequence of the reflection symmetries of the cube (Dutourand Deza, 2004). Figure 4 shows the tilings of the cube faces for the wrap-pings {4, 3+}b,c for small values of b and c. Given this simple parameteri-sation, it is easy to explore some basic properties of small examples withinthe three classes. Numerical experimentation gives the results reported inTarnai (2006) for the parameter s(b, c), the number of strands in the cubewrapping described by b and c. Note that s(b, c) = s(c, b), as exchange ofb and c simply flips the chirality of the wrapping. Further combinatorialinformation can be obtained from Dutour and Deza (2004) Table 6.

Strand counting for Class I is straightforward. A wrapping in Class Ihas either b = 0 or c = 0, and without loss of generality we take c = 0.The strands all lie parallel to the cube edges (Figure 2(a)), and the doublecover has Oh symmetry; each strand has length 4b, so s(b, 0) = 12(b2 +

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3,11,3

3,00,3

4,11,4

4,00,4

5,11,5

5,00,5

6,11,6

6,00,6

6,22,6

6,33,6

5,22,5

5,33,5

4,22,4

4,33,4

3,22,3

3,3

2,2

Figure 4: Tilings of the faces of the cube (adapted from Tarnai, 2006), fordiffering values of the parameter pair b, c. Wrappings belonging to ClassI appear (in two copies) along the horizonal axis; wrappings belonging toClass II appear along the central vertical axis; the two enantiomeric versionsof each Class III wrapping appear in mirror-symmetric off-axis positions.

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02)/4b = 3b. In Class II, the strands cross the faces at an angle of π/4 tothe edges (Figure 2(b)), and the length of each strand is three times that ofthe diagonal of a cube face, i.e., 6b. Hence s(b, b) = 12(b2+b2)/6b = 4b = 4c.

For counting strands in Class III, another useful observation is that forany pair b = kb0 and c = kc0 with k an integer, the number of strands scalesas s(b, c) = ks(b0, c0) = s(c, b), and so it is only necessary to understandthe behaviour of s(b, c) for the ‘canonical’ pairs where b and c are co-prime.Where b and c are co-prime, the number of strands is 3, 4 or 6, and thedouble covering has point-group symmetry D4, D3, or D2, with all strandsequivalent.

3 Graph-theory based approach

A more general perspective on the wrapping of cubes (and other polyhedra)comes from a graph theoretical approach. The geometric construction of theprevious section specifies a tiling (and so a wrapping) of the cube and hencefixes a polyhedral graph, T , where the faces are the square tiles, the edgesare tile edges, and the vertices are tile corners. Note that the vertices of theunderlying cube coincide with the 8 three-valent vertices of T , whereas theedges of the cube may run across faces and edges of T . The sum of anglesaround each 4-valent vertex of T is 2π, and the tiling is ‘locally flat’ at thesepoints; these vertices all lie on faces or edges of the underlying cube. Thecube is the convex realisation of T .

The dual of T , i.e., T ?, is obtained by placing a vertex at the centre ofeach square tile. The edges of T ? are then defined by adjacencies (sharededges) of square tiles. Row (iv) of Figure 2 shows the graphs T ? for threewrappings. As all faces of the primal graph (T ) are square, T ? is a 4-regulargraph. The faces of T ? are either quadrangular, or triangular (correspondingto the eight corners of the cube). This construction suggests the study of 4-regular polyhedral graphs as a basis for systematics of wrappings: T ? will bea general 4-regular polyhedral graph; its dual will be a tiling T correspondingto a wrapping of an underlying object.

The account below is closely based on the treatment of octahedritegraphs by Deza and co-workers (Deza and Shtogrin, 2003; Deza et al., 2003),and shows how their results can be applied to the wrapping problem forcubes and for more general polyhedra.

3.1 4-regular polyhedral graphs

A polyhedral graph (a graph that is planar and 3-connected (West, 2001))obeys the Euler theorem

v + f = e+ 2 (1)

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where v is the number of vertices, f the number of faces, and e the numberof edges. If the graph is 4-regular, e = 2v, and if fr is the number of faceswith r sides, we have, by counting faces and edges,∑

r

(4− r)fr = 8. (2)

An immediate consequence is that every 4-regular polyhedral graph hasf3 ≥ 8 triangular faces. An important subset of 4-regular polyhedra consistsof those with the minimum number of triangular faces, and with all otherfaces quadrangular, i.e., with f3 = 8 and f4 = 0 or f4 ≥ 2. These arethe polyhedra which are called octahedrites by Deza and Shtogrin (2003),and they are, in a sense, the equivalents amongst 4-regular polyhedra of thefullerenes (Fowler and Manolopoulos, 2006) amongst cubic polyhedra.

The numbers N of octahedrites with n vertices are known for smalln (A111361 in Sloane’s encyclopedia of integer sequences, (Sloane, 2008;Brinkmann et al., 2003)). They are N(n): 1(6); 0(7); 1(8); 1(9); 2(10);1(11); 5(12); 2(13); 8(14); 5(15); 12(16); 8(17); 25(18); 13(19); 30(20) . . ..The point groups allowed for octahedrites comprise the 18 possibilities Oh,O, D4d, D3d, D2d, D4h, D3h, D2h, D4, D3, D2, S4, C2v, C2h, C2, Cs, Ci,C1 (Deza et al., 2003). The subset of octahedrites that have octahedral (Oor Oh) symmetry is useful in describing wrappings of the cube; each is thedual, T ?, of a wrapping T .

Some definitions and facts about octahedrites are now briefly summarised.For details and proofs, the original papers by Deza and co-workers (Dezaand Shtogrin, 2003; Deza et al., 2003) should be consulted. Graphs of whichall vertices are of even degree are Eulerian, i.e., they admit circuits thatvisit every edge exactly once. Eulerian polyhedral graphs have no bridges(cut-edges) and no cut-vertices. For Eulerian graphs embedded in surfaces,we can define central circuits. A central circuit (CC) is constructed start-ing with a single edge, and visiting vertices according to the rule that thesequence enters and leaves any given vertex by opposite edges. For a finitegraph, this rule leads to a circuit. The relevance to the wrapping problemis that each CC of the 4-regular graph T ? corresponds to a strand in theweaving of T (with the CC being the mid-line of the strand).

Central circuits may be simple, or self-intersecting. The set of CC’spartitions the set of edges of the graph. In a 4-regular graph, the full setof CC’s provides a double cover of the vertices: every vertex is visited twiceby CC’s, either once each by two distinct CC’s, or twice by a single self-intersecting CC. The length of every CC is even, and the total length of allCC’s in the graph is 2n, by the double-cover property, with n the number ofvertices of the 4-regular graph (the number of squares in the primal). Thus,all strands are of even length, and, for a cube wrapping, their total lengthis 12S = 12(b2 + c2).

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A railroad is a circuit of square faces in an octahedrite. Octahedriteswithout railroads are called irreducible. In the context of cube wrappings,the duals of the wrappings with b and c co-prime are irreducible octahedrites.It can be proved that every irreducible octahedrite, of whatever symmetry,has at most six CC’s. Thus, for cube wrappings with b, c co-prime, thenumber of strands is limited to 3, 4, or 6, as noted earlier. There are onlyeight irreducible octahedrites in which all CC’s are simple circuits. Thevertex counts (and symmetries) are 6 (Oh), 12 (Oh, D3h), 14 (D4h), 20 (D2d),22 (C2v), 30 (O) and 32 (D4h) (Deza and Shtogrin, 2003). The three ofoctahedral symmetry correspond to cube wrappings with parameters {1, 0},{1, 1}, {2, 1}, i.e., one example from each of classes I, II and III.

Deza and co-workers also make an intriguing connection between octa-hedrites and knot theory. Every 4-valent plane graph can be seen as a regularalternating projection of an alternating knot or link (Kawauchi, 1996) andso a weaving is a physical manifestation of an alternating link. Since awrapping is equivalent to a 4-valent graph T ?, every wrapping correspondsto a weaving. Deza and Shtogrin (2003) catalogue the associations betweensome small octahedrites and well known objects of knot theory. Clearly, thiscould give an interesting direction for exploration in practical basketry.

Perhaps the most significant implication of the association between wrap-pings and octahedrites is that it soon becomes clear that there are manyother wrappable polyhedra beyond the simple cube. Any octahedrite T ? de-fines a tiling T . The Alexandrov existence and uniqueness theorems (Pak,2008) guarantee that T can be realised with all faces square as the wrappingof a unique underlying 8-vertex object P , where P is either a polyhedron ora ‘doubly covered polygon’, i.e., a degenerate polyhedron with just two faces.In general, many non-convex realisations are also possible, corresponding todifferent distributions of folds in the square faces. In fact, the volume of Pcan always be increased by taking a non-convex realisation (see Pak, 2008,Theorem 39.4). For practical construction of closed baskets, some of thesenon-convex realisations may, in fact, be preferable.

Figure 5 gives a complete catalogue of the octahedrites with n ≤ 16, andFigure 6 gives an example weaving derived from each of the six smallest oc-tahedrites. In each case we have chosen to show the weaving on the Alexan-drov polyhedron. A closed basket based on the smallest non-octahedrallysymmetric octahedrite (the square antiprism, 8-1 in Figure 5), and wovenby Felicity Wood, is shown in Figure 7.

It may be useful to recapitulate the relation between octahedrites ofoctahedral symmetry and wrappings of the cube. As Figure 2 row (iv)shows, each wrapping on the cube defines an octahedrite graph via the setof mid-lines of all strands. That octahedrite graph will have either fulloctahedral (Oh) or octahedral rotational (O) point-group symmetry. Con-versely, any octahedrite of Oh or O symmetry corresponds to a wrappingof the cube. The convex realisation of the dual of each such octahedrite is

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6-1 Oh 8-1† D4d 9-1† D3h 10-1 D4h 10-2 D2 11-1 C2v

12-1 Oh 12-2 D3h 12-3† D3d 12-4† D2 12-5 C2 13-1 C2v

13-2† C2 14-1 D4h 14-2 D4h 14-3 D2d 14-4† C2 14-5 D2

14-6 Cs 14-7 D2 14-8 C2 15-1 D3h 15-2† C2 15-3 Cs

15-4 Cs 15-5 C2 16-1 C2 16-2 D4d 16-3 D4h 16-4† D4d

16-5† D2 16-6† C2 16-7† C2 16-8† Cs 16-9† C2 16-10 Cs

16-11 D2 16-12 C1

Figure 5: A complete catalogue of octahedrites with N = 16 or fewer ver-tices. The labelling follows Deza and Shtogrin (2003), and includes thevertex number, isomer count and point group. The dagger added to thelabel indicates an octahedrite with only one central circuit, and hence onlyone strand in the weaving. The Schlegel diagrams are collated and redrawnfrom Deza and Shtogrin (2003).

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(a) (b) (c)

(d) (e) (f)

Figure 6: Examples of wrappings based on octahedrites. Each strand in awrapping is shown striped, with four grey stripes running along its length.The wrapped solids are the Alexandrov polyhedra derived from the six small-est octahedrites: (a) 6− 1 Oh; (b) 8− 1 D4d; (c) 9− 1 D3h; (d) 10− 1 D4h;(e) 10 − 2 D2; (f) 11 − 1 C2v. The Alexandrov polyhedra are combinatori-ally equivalent to: (a) the cube; (b) the square antiprism; (c) the bicappedtrigonal prism, which is also J14, the 14th Johnson solid (Johnson, 1966);(d) the cube; (e) the snub disphenoid or triangular dodecahedron, J84, (f)the gyrobifastigium, J26. In the numbering used in Read and Wilson (1998)these graphs are: (a) Tc46; (b) Tc249; (c) Tc168; (d) Tc46; (e) Tc301; (f)Tc94. In the numbering of 8-vertex coordination polyhedra given in Brittonand Dunitz (1973) they are labelled: (a) 257; (b) 128; (c) 198; (d) 257; (e)14; (f) 244.

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Figure 7: Weaving of the square antiprism. The ribbon forms a single closedstrand of 16 unit squares. This closed basket was constructed by FelicityWood, who also provided the photograph.

Figure 8: Wrappings of the cube. The tiling T in each case is the dual of anoctahedrite with octahedral rotational symmetry: 6 − 1 Oh; 12 − 1 Oh; 24Oh; 30 O; . . .. All four wrappings can be seen as inflations of the first (theunit cube) with (b, c) = (1, 0), (1, 1), (2, 0), (2, 1).

a decorated cube, with corners corresponding to triangular faces of the oc-tahedrite graph, and all 4-coordinate vertices of the octahedrite appearing,either on a cube edge, or in a flat region on a face. Figure 8 shows cubewrappings with (b, c) = (1, 0), (1, 1), (2, 0), (2, 1), which are derived from 6-,12-, 24-, and 30-vertex octahedrites, respectively.

Similar reasoning applies to the general, non-octahedrally symmetricoctahedrites. By the Goldberg-Coxeter construction (Dutour and Deza,2004; Deza and Dutour Sikiric, 2007), any octahedrite on n = n0 vertices canbe expanded to give an octahedrite on n = (b2+c2)n0 vertices. This involvesstretching and rotating the net so that each edge-length is multiplied by theinflation factor

√b2 + c2. Each octahedrite is therefore the parent of an

infinite sequence of inflated versions, and hence it is natural to define primeoctahedrites as those that are not produced by the inflation of any smaller

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Figure 9: Wrappings of the square antiprism where the triangular faces areright isosceles triangles. The octahedrites that generate these wrappingsare the first four Goldberg-Coxeter expansions of the octahedrite 8-1, thefirst octahedrite with non-octahedral symmetry, The four examples have(b, c) = (1, 0), (1, 1), (2, 0), (2, 1), where the pair of values (b, c) relates to theshorter edge of the triangles.

octahedrite. If the vertex count of a prime octahedrite is inflated by a factor(b2 + c2), where {b, c} = {b, 0} or {b, c} = {b, b}, the enlarged octahedritehas the same symmetries as the prime parent; otherwise it has at least the(proper) rotational symmetries. Each inflation of a prime octahedrite willhave a dual with a convex realisation, which will be identical to that ofthe dual of the prime parent, apart from a geometric scaling by

√b2 + c2.

Thus each family could be considered as a sequence of increasingly complexwrappings of the same underlying polyhedron P . Figure 8 shows the firstfour members of the family where the parent is the smallest octahedrite 6-1,and where P is the cube. Likewise, Figure 9 shows the first four members ofthe family where the parent is the next smallest octahedrite 8-1, and whereP is the square antiprism.

A natural question is: what convex polyhedra can be wrapped? For thosepolyhedra P that ultimately derive from octahedrites, we have a partialanswer. In this case, P must have 8 vertices, and the implication is thatthere are at most 258 combinatorially distinct P ; these comprise the 257 8-vertex polyhedra (Read and Wilson, 1998, chapter 5) and the doubly coveredoctagon. The examples in Figure 6 show that at least five of the set of 257polyhedra can be wrapped, some in multiple geometric realisations; the otherpolyhedral cases have not yet been explored, but we note that the octagonoccurs as P for the case 14-1, as shown in Figure 10.

4 Extensions

The family of ‘i-hedrites’ is a generalisation of the octahedrites: an i-hedritehas f2 = 8−i digonal faces and f3 = 8−2f2 triangular faces, with i = 4, . . . , 8

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(a) (b) (c)

Figure 10: Wrapping an octagon: (a) the octahedrite 14-1, redrawn fromFigure 5 to emphasise the full D4h symmetry (with arrows directed to avertex at infinity); (b) the tiling formed as the dual of the octahedrite; (c)one face of the wrapped polygon.

(Deza et al., 2003). Although not polyhedral, the duals of these graphs gen-erate a square tiling that is an intrinsically convex polyhedral surface, i.e.,it has everywhere non-negative curvature, and hence Alexandrov’s theoremstill applies. Duals of i-hedrites therefore generate wrappings of convex poly-hedra (or doubly-covered polygons) in much the same way as octahedrites.Figure 11 gives two examples of i-hedrites, their duals, and the wrappedobjects. It is even possible to go a step further in the generalisation of theoctahedrites, and allow faces of size 1 (loops) which give univalent verticesin the tiling.

Beyond the octahedrite and i-hedrite classes, those members of the widerclass of 4-regular polyhedra that have negative curvature also generate wrap-pings. The 4-regular polyhedra obey (2), i.e.,

1f3 − 0f4 − 1f5 − 2f6 − . . . = 8. (3)

Those with fr > 0 for some r > 4 have a region or regions of negative cur-vature. The numbers of general 4-regular polyhedra are given by sequenceA007022 in Sloane’s encyclopedia (Sloane, 2008; Brinkmann et al., 2003)).Figure 12 shows small examples based on dualising polyhedra with pentago-nal and hexagonal faces; in this case, non-convex realisations are inevitablebecause of the negative curvature, and so we have chosen symmetricallycrinkled structures to preserve the maximum Dnd symmetry of the under-lying polyhedra. Examples based on 4-regular polyhedra with even highersymmetries can also be constructed. An icosahedrally symmetric crinkledstructure is shown in Figure 13, similar in appearance to the 3-fold wovenobject presented in Plate D of Pedersen (1981).

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(a.i) (a.ii) (a.iii)

(b.i) (b.ii) (b.iii)

Figure 11: Examples of wrapping based on i-hedrites, showing (i) the i-hedrite graph, (ii) the dual, and (iii) the wrapped object, for (a) 10-1 D4

(with one vertex of the i-hedrite at infinity), and (b) 12-3 D2d (with onevertex of the dual at infinity) The numbering of the i-hedrites follows Dezaet al. (2003)).

15

(a.i) (a.ii) (a.iii)

(b.i) (b.ii) (b.iii)

(c.i) (c.ii) (c.iii)

Figure 12: A family of wrapped non-convex polyhedra. The panels show,in column (i), the primal polyhedra, the [n]-antiprisms with n = 4, 5, 6, i.e.,the square, pentagonal and hexagonal antiprisms with f3 = 2r, fr = 2 forr = 4, 5, 6, respectively; in column (ii), their duals, the [n]-trapezohedra; incolumn (iii), the wrapped objects. In the case of the [4]-antiprism (octa-hedrite 8-1), a convex realisation exists and is shown in Figure 9. For n ≥ 5,non-convexity makes crinkling unavoidable.

16

Figure 13: Wrapping of an icosahedrally symmetric non-convex polyhedron.The 4-regular polyhedral graph whose dual defines the tiling is the graph ofthe icosidodecahedron, with f5 = 12 and f3 = 20.

5 Conclusion

Starting from an analysis of wrappings of one simple highly-symmetric poly-hedron, the cube, it has been possible to identify infinite classes of potentialwrappings and closed baskets based on convex and non-convex polyhedra.Some open questions remain, such as the characterisation of the 8-vertexpolyhedra that may appear in wrappings derived from octahedrites: do allappear as Alexandrov polyhedra of wrappings, and if so, with what symme-try and geometrical realisation?

Weaving on intrinsically curved surfaces presents technical difficulties,e.g., the ‘draping’ problem in the manufacture of advanced composite com-ponents of complex geometry (Hancock and Potter, 2006). Extension ofthe present considerations, together with modern tow placement machines(Rudd et al., 1999) could help in the development of improved manufac-turing techniques. The present findings already provide a pattern-book forfuture artistic and practical endeavours.

Acknowledgements

PWF acknowledges support from the Royal Society/Wolfson Research MeritAward Scheme. TT and FK are grateful for financial support under OTKAgrant K81146. Prof. R. Connelly is thanked for a helpful discussion onconvexity. We thank M.A. Fowler and Dr A. Lengyel for help with pho-tographing the models, and Mrs F. Wood for making new weavings andallowing us to use photographs of her work.

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